What do 「ちょっと聞くけどさぁ and 」「ちょっとお聞きしますが」 mean? Do they mean

A. "I'd like to ask you something,"

or B. "I am going to ask you something"?

日本語を勉強している人は「would like to」とか「want to」みたいなことを言うために、「～たい」か「~たい〇〇と思う」という言葉を使うべきのことが教えられています。

How often do you actually use ～たい or ~たい〇〇と思う? Do you often leave them out?

Also is it common leave out the direct object(〇〇を) in Japanese?

To me, 「ちょっとお聞きしますが 」 looks like "I'm going to ask you a little." Not, "I'd like to ask you something." There doesn't seem to be a direct object in this sentence. In other words, there is nothing to mark with を.

OK, I'll explain how I feel ちょっとお聞きしたいのですが(A) and ちょっとお聞きしますが(B)　respectively.
I told you A sounds a bit more polite and milder than B. That's because A just states what you want to do, I mean it's up to the hearer whether they answer or not, while B sounds like you're going to do that and the hearer is supposed to answer. In other words B may sound a bit pushy, though it depends on the tone of the voice and the case.

When you guestion intensely someone, B sounds perfectly natural, though I wouldn't say A is unnatural. Say B is 100 points then A is 97 points.
(B)ちょっとお聞きしますが、次の電車はいつ来るんですか？
(A)ちょっとお聞きしたいのですが、次の電車はいつ来るんですか？
(situation:trains are off-shcedule and you're annoyed with that. "When the next train will come?" to a station attendant)

But it doesn't mean you can use B only when you grill someone.
(B)ちょっとお聞きしますが、ここは撮影禁止ですか？
(A)ちょっとお聞きしたいのですが、ここは撮影禁止ですか？
(Is taking a picture prohibited here?)
A sounds just a little bit more polite than B, but they sound almost the same to me. The way how you say it (voice and facial expression) is far more important.

conclusion: A sounds a bit more polite than B, but there's no big difference.